Direction Cosines and Ratios
Direction cosines (l, m, n) are the cosines of the angles a line makes with the X, Y, and Z axes, satisfying l^2 + m^2 + n^2 = 1. Direction ratios (a, b, c) are any numbers proportional to these cosines, representing the vector direction of the line.
Equation of a Line
A line in 3D is defined by a point it passes through and its direction vector. The Cartesian form is (x - x1)/a = (y - y1)/b = (z - z1)/c, where (x1, y1, z1) is the point and (a, b, c) are the direction ratios.
Equation of a Plane
A plane is defined by a point on it and a vector perpendicular to it (the normal vector). The general equation is Ax + By + Cz + D = 0, where (A, B, C) represents the direction ratios of the normal vector.
Angle Between Line and Plane
Unlike the angle between two lines or two planes which uses cosine, the angle theta between a line and a plane uses sine. The formula is sin(theta) = |a*A + b*B + c*C| / (sqrt(a^2 + b^2 + c^2) * sqrt(A^2 + B^2 + C^2)).
⚡ What to Skip
If the exam is just two weeks away, you can safely skip the complex derivations of the shortest distance between skew lines in vector form. Just memorize the final Cartesian formula and focus on basic plane equations and point-to-plane distances.
🏆 Exam Strategy
First, always check if the given options can be eliminated by substituting points directly into the equations. Second, pay close attention to the sign of the constant term D when using distance formulas. Third, remember that the angle between a line and a plane uses the sine function, not cosine.